Such a large deviation result is valid for mixing enough dynamics
and observables with probability distribution function that decay sufficiently fast for large values of the observable. For a Gaussian process with exponential correlation function, the autocorrelation time and the mixing time are of the same order of magnitude. However, in more complex dynamics the picture can be more complicated.
Sufficient conditions are given by Donsker-Varadhan's
theory for Markov processes [1]}, [2]} or by [3]}, [4]} for
dynamical systems. The more general result \(\rho (a,T)\underset{T\rightarrow \infty }{\sim }C(a,T\text{)}\mbox{e}^{-TI(a)}\) 
would imply \(\rho (a,T)\underset{T\rightarrow \infty }{\asymp }\mbox{e}^{-TI(a)}\) 
if \(C(a,T)\)  increases less than exponentially for large \(T\) . The
logarithmic equivalence thus means that the prefactor \(C\)  is subdominant
in the large \(T\)  limit with respect to the the exponential term,
and is not determined.
